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Pleiotropy, Sex-Dependent Allelic Effects and GE Interactions

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Pleiotropy, Sex-Dependent Allelic Effects and GE Interactions Copyright 2004 by the Genetics Society of America Polygenic Variation Maintained by Balancing Selection: Pleiotropy, Sex-Dependent Allelic Effects and G ϫ E Interactions Michael Turelli*,1 and N. H. Barton† *Section of Evolution and Ecology and Center for Population Biology, University of California, Davis, California 95616 and †Institute of Cell, Animal and Population Biology, University of Edinburgh, Edinburgh EH9 3JT, United Kingdom Manuscript received May 29, 2003 Accepted for publication October 17, 2003 ABSTRACT We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G ϫ E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model pro- duces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleio- tropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G ϫ E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The criti- cal value depends on the correlation between substitution effects at different loci. For large positive ␳2 Ͼ correlations (e.g., ij 3/4), even extreme fluctuations in allelic effects cannot maintain variation. Surpris- ingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms. T remains a challenge for evolutionary geneticists to lelic effects (treating spatial and temporal heterogene- I understand the additive genetic variance observed ity separately), and sex-dependent allelic effects. The for most traits in most populations. Given the ubiquity thread that unites these scenarios is that, under weak of additive genetic variation, it is natural to seek an selection, each produces very similar allele-frequency explanation in terms of ubiquitous forces. Lande (1975) dynamics and polymorphism conditions. An empirical proposed mutation-selection balance. However, over motivation for these analyses is that alleles of intermedi- the past 25 years, attempts to explain standing levels ate frequency seem to contribute to phenotypic varia- of quantitative genetic variation in terms of mutation- tion in natural populations (e.g., Mackay and Langley selection balance have been at best only partially success- 1990; Long et al. 2000). Such polymorphisms are incom- ful (e.g., Caballero and Keightley 1994; Charles- patible with mutation-selection balance for plausible worth and Hughes 2000; but see Zhang and Hill levels of selection and mutation. 2002). One alternative is that some form of balancing The mathematical motivation for our analyses is selection, unconnected to the trait of interest, may ac- Wright’s (1935) demonstration that stabilizing selec- count for persistent polymorphism at the underlying tion tends to eliminate polygenic variation. Using a weak- loci (e.g., Robertson 1965; Bulmer 1973; Gillespie selection approximation, he showed that at most one 1984; Barton 1990). In contrast to such pleiotropic ex- locus is expected to remain polymorphic at equilibrium. planations, balancing selection might arise from varia- More recent analyses of strong selection (Nagylaki tion in the effects of alleles that contribute to the trait, 1989; Bu¨rger and Gimelfarb 1999) have found that for instance, through genotype-by-environment (G ϫ E) two-locus polymorphisms can be stably maintained with interactions. Here we explore four scenarios in which sufficiently strong selection and sufficient interlocus variance-depleting stabilizing selection interacts with pleio- variation in allelic effects. We provide new simulations tropic balancing selection, environment-dependent al- that further illustrate the restrictive conditions needed to maintain even stable two-locus polymorphisms for ad- ditive traits under stabilizing selection and loose linkage. 1Corresponding author: Section of Evolution and Ecology, University of California, 1 Shields Ave., Davis, CA 95616. Robertson (1965) proposed that additive variation E-mail: [email protected] may be maintained by pleiotropically induced overdom- Genetics 166: 1053–1079 ( February 2004) 1054 M. Turelli and N. H. Barton inant selection, which counteracts the effects of stabiliz- that sex-dependent allelic effects do not stably maintain ing selection. His conjecture was explored analytically polygenic variation for additive traits. by Bulmer (1973) for diallelic loci and extended to All of our analyses assume that selection is weak multiple alleles by Gillespie (1984). Both assumed enough relative to recombination that linkage disequi- equal allelic effects across loci, symmetric overdomi- librium is negligible. We also assume diallelic loci. It is nance of equal intensity at all loci, and that the popula- not clear to us how restrictive this assumption is. In tion mean at equilibrium coincided with the optimal models of mutation-selection balance, two-allele and trait value. They found lower bounds on the intensity continuum-of-allele models give similar results provided of overdominant selection required to maintain stable that the alleles responsible for variation are rare (Tur- multilocus polymorphisms. Our diallelic analyses gener- elli 1984; Slatkin and Frank 1990). However, when alize theirs and that of Zhivotovsky and Gavrilets loci are highly polymorphic (as might occur under bal- (1992), by allowing for unequal allelic effects and arbi- ancing selection), continuum-of-allele models can give trary overdominance across loci and by considering the qualitatively different results (Bu¨rger 1999; Waxman simultaneous stability of alternative equilibria at which and Peck 1999; Waxman 2003). Nevertheless, we be- the population mean can depart from the optimum. lieve that models with two alleles are a better approxi- Gillespie and Turelli (1989) showed how balancing mation to reality (where there will usually be a few selection could arise at individual loci by averaging over discrete alleles) than are a continuum of alleles, particu- randomly fluctuating allelic effects. In their symmetric larly when pleiotropy is considered, because it takes an model of G ϫ E interactions, all alleles have essentially extraordinary number of discrete alleles to approxi- the same mean and variance of effects. With this ex- mate even a two-dimensional continuum (Turelli 1985; treme symmetry assumption, even slight fluctuations can Wagner 1989). Models with discrete alleles also pre- maintain indefinitely many alleles at an arbitrary num- clude a particular multilocus genotype that produces ber of loci. However, the essential interchangeability the highest fitness under all conditions. By implicitly allowing such genotypes, Via and Lande (1987) con- of the alleles implies that there will be essentially no ϫ correlation between the phenotypes produced by a cluded that G E interactions could not maintain sta- given genotype across unrelated environments (i.e., two ble polygenic variation. Another critical assumption is that the temporal and environments chosen at random from the distribution spatial scales of fluctuating allelic effects are sufficiently of environments responsible for maintaining variation; small, relative to the timescale of selection, that we can Gillespie and Turelli 1989, 1990; Gimelfarb 1990). average over these fluctuations to approximate the al- Genetic variation that shows so little consistency of ef- lele-frequency dynamics with deterministic differential fects would severely limit the resemblance between par- equations. Both the linkage equilibrium and averaging ents and offspring across different environments. approximations were made by Gillespie and Turelli Below we explore the consequences of allowing ap- (1989), and we explore their validity numerically with preciable differences in the mean effects of different temporal fluctuations in allelic effects (and sex-depen- alleles. We show that under simple forms of spatial and dent allelic effects). We conjecture that more highly temporal variation in allelic effects, the conditions for autocorrelated temporal fluctuations would maintain the maintenance of variation become much more re- less variation (Gillespie and Guess 1978), whereas a strictive than those indicated by Gillespie and Turelli coarser spatial variation can maintain more variation (1989). Nevertheless, a surprisingly simple necessary (Barton and Turelli 1989; Barton 1999). condition for the maintenance of variation emerges. Our analyses show that balancing selection can main- Our weak-selection approximations apply to a broad tain variation at loci for which the intensity of balancing class of selection
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